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                    线性模型
            
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                    集成学习
            
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                    计算学习理论
            
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                                <h1 id="&#x897F;&#x74DC;&#x4E66;&#x6982;&#x5FF5;">&#x897F;&#x74DC;&#x4E66;&#x6982;&#x5FF5;</h1>
<h2 id="&#x7B2C;12&#x7AE0;-&#x8BA1;&#x7B97;&#x5B66;&#x4E60;&#x7406;&#x8BBA;">&#x7B2C;12&#x7AE0; &#x8BA1;&#x7B97;&#x5B66;&#x4E60;&#x7406;&#x8BBA;</h2>
<ul>
<li><p>Page267: &#x8BA1;&#x7B97;&#x5B66;&#x4E60;&#x7406;&#x8BBA;&#xFF08;computational learning theory&#xFF09;</p>
<p>  &#x8BA1;&#x7B97;&#x5B66;&#x4E60;&#x7406;&#x8BBA;&#x7814;&#x7A76;&#x7684;&#x662F;&#x5173;&#x4E8E;&#x901A;&#x8FC7;&#x201C;&#x8BA1;&#x7B97;&#x201D;&#x6765;&#x8FDB;&#x884C;&#x201C;&#x5B66;&#x4E60;&#x201D;&#x7684;&#x7406;&#x8BBA;&#xFF0C;&#x5373;&#x5173;&#x4E8E;&#x673A;&#x5668;&#x5B66;&#x4E60;&#x7684;&#x7406;&#x8BBA;&#x57FA;&#x7840;&#xFF0C;&#x5176;&#x76EE;&#x7684;&#x662F;&#x5206;&#x6790;&#x5B66;&#x4E60;&#x4EFB;&#x52A1;&#x7684;&#x56F0;&#x96BE;&#x672C;&#x8D28;&#xFF0C;&#x4E3A;&#x5B66;&#x4E60;&#x7B97;&#x6CD5;&#x63D0;&#x4F9B;&#x7406;&#x8BBA;&#x4FDD;&#x8BC1;&#xFF0C;&#x5E76;&#x6839;&#x636E;&#x5206;&#x6790;&#x7ED3;&#x679C;&#x6307;&#x5BFC;&#x7B97;&#x6CD5;&#x8BBE;&#x8BA1;&#x3002;</p>
</li>
<li><p>Page268: Jensen&#x4E0D;&#x7B49;&#x5F0F;</p>
<p>  <script type="math/tex; ">f(\mathbb{E}(x)) \leqq  \mathbb{E}(f(x))</script></p>
<p>  &#x671F;&#x671B;&#x7684;&#x51FD;&#x6570;&#x5C0F;&#x4E8E;&#x51FD;&#x6570;&#x7684;&#x671F;&#x671B;</p>
</li>
<li><p>Page268: Hoeffding&#x4E0D;&#x7B49;&#x5F0F;</p>
<p>  &#x82E5;<script type="math/tex; ">x_1,x_2,...,x_m</script>&#x4E3A;<script type="math/tex; ">m</script>&#x4E2A;&#x72EC;&#x7ACB;&#x968F;&#x673A;&#x53D8;&#x91CF;&#xFF0C;&#x4E14;&#x6EE1;&#x8DB3;<script type="math/tex; ">0\leqq x_i\leqq 1</script>&#xFF0C;&#x5219;&#x5BF9;&#x4E8E;&#x4EFB;&#x610F;<script type="math/tex; ">\epsilon\geqq 0</script>&#x6709;&#xFF1A;
  <script type="math/tex; ">P(\frac{1}{m}\sum_{i=1}^{m}x_i-\frac{1}{m}\sum_{i=1}^m\mathbb{E}(x_i)\geqq \epsilon)\leqq exp(-2m\epsilon^2)</script>
  <script type="math/tex; ">P(\mid \frac{1}{m}\sum_{i=1}^{m}x_i-\frac{1}{m}\sum_{i=1}^m\mathbb{E}(x_i)\mid \geqq \epsilon)\leqq 2exp(-2m\epsilon^2)</script></p>
</li>
<li><p>Page268: McDiarmid &#x4E0D;&#x7B49;&#x5F0F;</p>
<p>  &#x82E5;<script type="math/tex; ">x_1,x_2,...,x_m</script>&#x4E3A;<script type="math/tex; ">m</script>&#x4E2A;&#x72EC;&#x7ACB;&#x968F;&#x673A;&#x53D8;&#x91CF;&#xFF0C;&#x4E14;&#x5BF9;&#x4EFB;&#x610F;<script type="math/tex; ">1\leqq i\leqq m</script>&#xFF0C;&#x51FD;&#x6570;<script type="math/tex; ">f</script>&#x6EE1;&#x8DB3;</p>
<p>  <script type="math/tex; ">\sum_{x_1,x_2,...,x_m, x^{'}_i } |f(x_1,...,x_m) - f(x_1, ... , x_{i-1}, x^{'}_i, x_{i+1}, ..., x_m)| \leqq c_i</script>&#xFF0C;</p>
<p>  &#x5219;&#x5BF9;&#x4EFB;&#x610F;<script type="math/tex; ">\epsilon\geqq 0</script>&#xFF0C;&#x6709;
  <script type="math/tex; ">P(f(x_1,...,x_m)-E(f(x_1,...,x_m))\geqq \epsilon)\leqq exp(\frac{-2\epsilon^2}{\sum_i c_i^2})</script>
  <script type="math/tex; ">P(\mid f(x_1,...,x_m)-E(f(x_1,...,x_m))\mid \geqq \epsilon)\leqq exp(\frac{-2\epsilon^2}{\sum_i c_i^2})</script></p>
</li>
<li>Page268: &#x6982;&#x7387;&#x8FD1;&#x4F3C;&#x6B63;&#x786E;</li>
<li><p>Page268: &#x6982;&#x5FF5;&#x7C7B;(concept class)</p>
<p>  &#x4EE4;c&#x8868;&#x793A;&#x6982;&#x5FF5;&#xFF0C;&#x8FD9;&#x662F;&#x4ECE;&#x6837;&#x672C;&#x7A7A;&#x95F4;<script type="math/tex; ">\mathcal{X}</script>&#x5230;&#x6807;&#x8BB0;&#x7A7A;&#x95F4;<script type="math/tex; ">\mathcal{Y}</script>&#x7684;&#x6620;&#x5C04;&#xFF0C;&#x5B83;&#x51B3;&#x5B9A;&#x793A;&#x4F8B;x&#x7684;&#x771F;&#x5B9E;&#x6807;&#x8BB0;y&#xFF0C;&#x82E5;&#x5BF9;&#x4EFB;&#x4F55;&#x6837;&#x4F8B;(x,y)&#x6709;c(x)=y&#x6210;&#x7ACB;&#xFF0C;&#x5219;&#x79F0;c&#x4E3A;&#x76EE;&#x6807;&#x6982;&#x5FF5;&#xFF0C;&#x6240;&#x6709;&#x6211;&#x4EEC;&#x5E0C;&#x671B;&#x5B66;&#x5F97;&#x7684;&#x76EE;&#x6807;&#x6982;&#x5FF5;&#x6240;&#x6784;&#x6210;&#x7684;&#x96C6;&#x5408;&#x79F0;&#x4E3A;&#x6982;&#x5FF5;&#x7C7B;&#xFF0C;&#x7528;&#x7B26;&#x53F7;<script type="math/tex; ">\mathcal{C}</script>&#x8868;&#x793A;&#x3002;</p>
</li>
<li><p>Page268: &#x5047;&#x8BBE;&#x7A7A;&#x95F4;&#xFF08;hypothesis space&#xFF09;</p>
</li>
</ul>
<p>&#x7ED9;&#x5B9A;&#x5B66;&#x4E60;&#x7B97;&#x6CD5;<script type="math/tex; ">\mathfrak{L}</script>&#xFF0C;&#x5B83;&#x6240;&#x8003;&#x8651;&#x7684;&#x6240;&#x6709;&#x53EF;&#x80FD;&#x6982;&#x5FF5;&#x7684;&#x96C6;&#x5408;&#x79F0;&#x4E3A;&#x201C;&#x5047;&#x8BBE;&#x7A7A;&#x95F4;&#x201D;&#xFF0C;&#x7528;&#x7B26;&#x53F7;<script type="math/tex; ">\mathcal{H}</script>&#x8868;&#x793A;&#x3002;&#x7531;&#x4E8E;&#x5B66;&#x4E60;&#x7B97;&#x6CD5;&#x4E8B;&#x5148;&#x5E76;&#x4E0D;&#x77E5;&#x9053;&#x6982;&#x5FF5;&#x7C7B;&#x7684;&#x771F;&#x5B9E;&#x5B58;&#x5728;&#xFF0C;&#x56E0;&#x6B64;&#x5047;&#x8BBE;&#x7A7A;&#x95F4;&#x548C;&#x6982;&#x5FF5;&#x7C7B;&#x5F80;&#x5F80;&#x4E0D;&#x540C;&#x3002;</p>
<ul>
<li><p>Page269: PAC&#x8FA8;&#x8BC6;(PAC Identify)</p>
<p>  &#x5BF9;&#x4E8E;<script type="math/tex; ">0<\epsilon, \delta<1</script>&#xFF0C;&#x6240;&#x6709;<script type="math/tex; ">c\in\mathcal{C}</script>&#x548C;&#x5206;&#x5E03;<script type="math/tex; ">\mathcal{D}</script>&#xFF0C;&#x82E5;&#x5B58;&#x5728;&#x5B66;&#x4E60;&#x7B97;&#x6CD5;<script type="math/tex; ">\mathfrak{L}</script>&#xFF0C;&#x5176;&#x8F93;&#x51FA;&#x5047;&#x8BBE;<script type="math/tex; ">h\in\mathcal{H}</script>&#x6EE1;&#x8DB3;</p>
<p>  <script type="math/tex; ">P(E(h)\leqq \epsilon)\geqq 1-\delta</script></p>
<p>  &#x5219;&#x79F0;&#x5B66;&#x4E60;&#x7B97;&#x6CD5;<script type="math/tex; ">\mathfrak{L}</script>&#x80FD;&#x4ECE;&#x5047;&#x8BBE;&#x7A7A;&#x95F4;<script type="math/tex; ">\mathcal{H}</script>&#x4E2D;PAC&#x8FA8;&#x8BC6;&#x6982;&#x5FF5;&#x7C7B;<script type="math/tex; ">\mathcal{C}</script>&#xFF0C;&#x8FD9;&#x6837;&#x7684;&#x5B66;&#x4E60;&#x7B97;&#x6CD5;<script type="math/tex; ">\mathfrak{L}</script>&#x80FD;&#x4ECE;&#x5047;&#x8BBE;&#x7A7A;&#x95F4;<script type="math/tex; ">\mathcal{H}</script>&#x4E2D;PAC&#x8FA8;&#x8BC6;&#x6982;&#x5FF5;&#x7C7B;<script type="math/tex; ">\mathcal{C}</script>.</p>
</li>
<li><p>Page269: PAC&#x53EF;&#x5B66;&#x4E60;(PAC Learnable)</p>
<p>  &#x4EE4;<script type="math/tex; ">m</script>&#x8868;&#x793A;&#x4ECE;&#x5206;&#x5E03;<script type="math/tex; ">\mathcal{D}</script>&#x4E2D;&#x72EC;&#x7ACB;&#x540C;&#x5206;&#x5E03;&#x91C7;&#x6837;&#x5F97;&#x5230;&#x7684;&#x6837;&#x4F8B;&#x6570;&#x76EE;&#xFF0C;<script type="math/tex; ">0<\epsilon,\delta<1</script>&#xFF0C;&#x5BF9;&#x6240;&#x6709;&#x5206;&#x5E03;<script type="math/tex; ">\mathcal{D}</script>&#xFF0C;&#x82E5;&#x5B58;&#x5728;&#x5B66;&#x4E60;&#x7B97;&#x6CD5;<script type="math/tex; ">\mathfrak{L}</script>&#x548C;&#x591A;&#x9879;&#x5F0F;&#x51FD;&#x6570;<script type="math/tex; ">poly(.,.,.,.)</script>&#xFF0C;&#x4F7F;&#x5F97;&#x5BF9;&#x4E8E;&#x4EFB;&#x4F55;<script type="math/tex; ">m\geqq poly(1/\epsilon,1/\delta,size(x),size(c))</script>, <script type="math/tex; ">\mathfrak{L}</script>&#x80FD;&#x4ECE;&#x5047;&#x8BBE;&#x7A7A;&#x95F4;<script type="math/tex; ">\mathcal{H}</script>&#x4E2D;PAC&#x8FA8;&#x8BC6;&#x6982;&#x5FF5;&#x7C7B;<script type="math/tex; ">\mathcal{C}</script>&#xFF0C;&#x5219;&#x79F0;&#x6982;&#x5FF5;&#x7C7B;<script type="math/tex; ">\mathcal{C}</script>&#x5BF9;&#x5047;&#x8BBE;&#x7A7A;&#x95F4;<script type="math/tex; ">\mathcal{H}</script>&#x800C;&#x8A00;&#x662F;PAC&#x53EF;&#x5B66;&#x4E60;&#x7684;&#xFF0C;&#x4E5F;&#x7B80;&#x79F0;&#x6982;&#x5FF5;&#x7C7B;<script type="math/tex; ">\mathcal{C}</script>&#x662F;PAC&#x53EF;&#x5B66;&#x4E60;&#x7684;&#x3002;&#x5BF9;&#x4E8E;&#x8BA1;&#x7B97;&#x673A;&#x7B97;&#x6CD5;&#x6765;&#x8BF4;&#xFF0C;&#x5FC5;&#x987B;&#x8003;&#x8651;&#x65F6;&#x95F4;&#x590D;&#x6742;&#x5EA6;&#xFF0C;&#x4E8E;&#x662F;&#xFF1A;</p>
</li>
<li><p>Page270: PAC&#x5B66;&#x4E60;&#x7B97;&#x6CD5;&#xFF08;PAC Learning Algorithm&#xFF09;</p>
<p>  &#x82E5;&#x5B66;&#x4E60;&#x7B97;&#x6CD5;<script type="math/tex; ">\mathfrak{L}</script>&#x4F7F;&#x6982;&#x5FF5;&#x7C7B;<script type="math/tex; ">\mathcal{C}</script>&#x4E3A;PAC&#x53EF;&#x5B66;&#x4E60;&#x7684;&#xFF0C;&#x4E14;<script type="math/tex; ">\mathfrak{L}</script>&#x7684;&#x8FD0;&#x884C;&#x65F6;&#x95F4;&#x4E5F;&#x662F;&#x591A;&#x9879;&#x5F0F;&#x51FD;&#x6570;<script type="math/tex; ">poly(1/\epsilon,1/\delta,size(x),size(c))</script>&#xFF0C;&#x5219;&#x79F0;&#x6982;&#x5FF5;&#x7C7B;<script type="math/tex; ">\mathcal{C}</script>&#x662F;&#x9AD8;&#x6548;PAC&#x53EF;&#x5B66;&#x4E60;&#x7684;&#xFF0C;&#x79F0;<script type="math/tex; ">\mathfrak{L}</script>&#x4E3A;&#x6982;&#x5FF5;&#x7C7B;<script type="math/tex; ">\mathcal{C}</script>&#x7684;PAC&#x5B66;&#x4E60;&#x7B97;&#x6CD5;&#x3002;</p>
</li>
<li><p>Page269: &#x65F6;&#x95F4;&#x590D;&#x6742;&#x5EA6;</p>
<p>  &#x5047;&#x5B9A;&#x5B66;&#x4E60;&#x7B97;&#x6CD5;<script type="math/tex; ">\mathfrak{L}</script>&#x5904;&#x7406;&#x6BCF;&#x4E2A;&#x6837;&#x672C;&#x7684;&#x65F6;&#x95F4;&#x4E3A;&#x5E38;&#x6570;&#xFF0C;&#x5219;<script type="math/tex; ">\mathfrak{L}</script>&#x7684;&#x65F6;&#x95F4;&#x590D;&#x6742;&#x5EA6;&#x7B49;&#x4EF7;&#x4E8E;&#x6837;&#x672C;&#x590D;&#x6742;&#x5EA6;&#x3002;</p>
</li>
<li><p>Page270: &#x6837;&#x672C;&#x590D;&#x6742;&#x5EA6;</p>
<p>  &#x6EE1;&#x8DB3;PAC&#x5B66;&#x4E60;&#x7B97;&#x6CD5;<script type="math/tex; ">\mathfrak{L}</script>&#x6240;&#x9700;&#x7684;<script type="math/tex; ">m\geqq  poly(1/\epsilon,1/\delta,size(x),size(c))</script>&#x4E2D;&#x6700;&#x5C0F;&#x7684;m&#xFF0C;&#x79F0;&#x4E3A;&#x5B66;&#x4E60;&#x7B97;&#x6CD5;<script type="math/tex; ">\mathfrak{L}</script>&#x7684;&#x6837;&#x672C;&#x590D;&#x6742;&#x5EA6;&#x3002;</p>
</li>
<li><p>Page269: &#x4E0D;&#x53EF;&#x5206;(272)(non-seperable)</p>
<p>  &#x5BF9;&#x4E8E;&#x8F83;&#x4E3A;&#x56F0;&#x96BE;&#x7684;&#x5B66;&#x4E60;&#x95EE;&#x9898;&#xFF0C;&#x76EE;&#x6807;&#x6982;&#x5FF5;<script type="math/tex; ">c</script>&#x5F80;&#x5F80;&#x4E0D;&#x5B58;&#x5728;&#x4E8E;&#x5047;&#x8BBE;&#x7A7A;&#x95F4;<script type="math/tex; ">\mathcal{H}</script>&#x4E2D;&#xFF0C;&#x5047;&#x5B9A;&#x5BF9;&#x4E8E;&#x4EFB;&#x4F55;<script type="math/tex; ">h\in\mathcal{H}, \hat{E(h)}\neq0</script>&#xFF0C;&#x4E5F;&#x5C31;&#x662F;&#x8BF4;&#xFF0C;<script type="math/tex; ">\mathcal{H}</script>&#x4E2D;&#x7684;&#x4EFB;&#x610F;&#x4E00;&#x4E2A;&#x5047;&#x8BBE;&#x90FD;&#x4F1A;&#x5728;&#x8BAD;&#x7EC3;&#x96C6;&#x4E0A;&#x51FA;&#x73B0;&#x6216;&#x591A;&#x6216;&#x5C11;&#x7684;&#x9519;&#x8BEF;&#x3002;<script type="math/tex; ">\mathcal{H}</script>&#x4E2D;&#x4E0D;&#x5B58;&#x5728;&#x4EFB;&#x4F55;&#x5047;&#x8BBE;&#x80FD;&#x5C06;&#x6240;&#x6709;&#x793A;&#x4F8B;&#x5B8C;&#x5168;&#x6B63;&#x786E;&#x5206;&#x5F00;&#xFF0C;&#x5219;&#x79F0;&#x8BE5;&#x95EE;&#x9898;&#x5BF9;&#x4E8E;&#x5B66;&#x4E60;&#x7B97;&#x6CD5;<script type="math/tex; ">\mathfrak{L}</script>&#x662F;&#x4E0D;&#x53EF;&#x5206;&#x7684;&#xFF0C;&#x4EA6;&#x79F0;&#x4E0D;&#x4E00;&#x81F4;&#x7684;&#x3002;</p>
</li>
<li><p>Page269: &#x4E0D;&#x4E00;&#x81F4;(non-consistent)</p>
<p>  &#x5373;&#x4E0D;&#x53EF;&#x5206;&#x3002;</p>
</li>
<li><p>Page269: &#x53EF;&#x5206;(270)(seperable)</p>
<p>   &#x53EF;&#x5206;&#x60C5;&#x5F62;&#x610F;&#x5473;&#x7740;&#x76EE;&#x6807;&#x6982;&#x5FF5;<script type="math/tex; ">c</script>&#x5C5E;&#x4E8E;&#x5047;&#x8BBE;&#x7A7A;&#x95F4;<script type="math/tex; ">\mathcal{H}</script>&#xFF0C; &#x5373; <script type="math/tex; ">c\in\mathcal{H}</script>&#xFF0C;<script type="math/tex; ">\mathcal{H}</script>&#x4E2D;&#x5B58;&#x5728;&#x5047;&#x8BBE;&#x80FD;&#x5C06;&#x6240;&#x6709;&#x793A;&#x4F8B;&#x6309;&#x4E0E;&#x771F;&#x5B9E;&#x6807;&#x8BB0;&#x4E00;&#x81F4;&#x7684;&#x65B9;&#x5F0F;&#x5B8C;&#x5168;&#x5206;&#x5F00;&#xFF0C;&#x6211;&#x4EEC;&#x5C06;&#x8BE5;&#x95EE;&#x9898;&#x5BF9;&#x5B66;&#x4E60;&#x7B97;&#x6CD5;<script type="math/tex; ">\mathfrak{L}</script>&#x662F;&#x53EF;&#x5206;&#x7684;&#xFF0C;&#x4EA6;&#x79F0;&#x4E00;&#x81F4;&#x7684;&#x3002;</p>
</li>
<li><p>Page270: &#x6070;PAC&#x53EF;&#x5B66;&#x4E60;(properly PAC learnable)</p>
<p>  &#x82E5;&#x5B66;&#x4E60;&#x7B97;&#x6CD5;<script type="math/tex; ">\mathfrak{L}</script>&#x4F7F;&#x6982;&#x5FF5;&#x7C7B;<script type="math/tex; ">\mathcal{C}</script>&#x4E3A;PAC&#x53EF;&#x5B66;&#x4E60;&#x7684;&#xFF0C;&#x4E14;<script type="math/tex; ">\mathfrak{L}</script>&#x7684;&#x8FD0;&#x884C;&#x65F6;&#x95F4;&#x4E5F;&#x662F;&#x591A;&#x9879;&#x5F0F;&#x51FD;&#x6570;<script type="math/tex; ">poly(1/\epsilon, 1/\delta, size(x), size(c))</script>&#xFF0C;&#x5219;&#x79F0;&#x6982;&#x5FF5;&#x7C7B;<script type="math/tex; ">\mathcal{C}</script>&#x662F;&#x9AD8;&#x6548;PAC&#x53EF;&#x5B66;&#x4E60;&#x7684;&#xFF0C;&#x79F0;<script type="math/tex; ">\mathfrak{L}</script>&#x4E3A;&#x6982;&#x5FF5;&#x7C7B;<script type="math/tex; ">\mathcal{C}</script>&#x7684;PAC&#x5B66;&#x4E60;&#x7B97;&#x6CD5;&#x3002;&#x82E5;&#x5728;PAC&#x5B66;&#x4E60;&#x4E2D;&#x5047;&#x8BBE;&#x7A7A;&#x95F4;&#x4E0E;&#x6982;&#x5FF5;&#x7C7B;&#x5B8C;&#x5168;&#x76F8;&#x540C;&#xFF0C;&#x79F0;&#x4E3A;&#x6070;PAC&#x53EF;&#x5B66;&#x4E60;&#xFF0C;&#x8FD9;&#x610F;&#x5473;&#x7740;&#x5B66;&#x4E60;&#x7B97;&#x6CD5;&#x7684;&#x80FD;&#x529B;&#x4E0E;&#x5B66;&#x4E60;&#x4EFB;&#x52A1;&#x6070;&#x597D;&#x5339;&#x914D;&#x3002;</p>
</li>
<li><p>Page270: &#x6709;&#x9650;&#x5047;&#x8BBE;&#x7A7A;&#x95F4;</p>
<p>  &#x4E00;&#x822C;&#x800C;&#x8A00;&#xFF0C;<script type="math/tex; ">\mathcal{H}</script>&#x8D8A;&#x5927;&#xFF0C;&#x5176;&#x5305;&#x542B;&#x4EFB;&#x610F;&#x76EE;&#x6807;&#x6982;&#x5FF5;&#x7684;&#x53EF;&#x80FD;&#x6027;&#x8D8A;&#x5927;&#xFF0C;&#x4F46;&#x4ECE;&#x4E2D;&#x627E;&#x5230;&#x67D0;&#x4E2A;&#x5177;&#x4F53;&#x6982;&#x5FF5;&#x7684;&#x96BE;&#x5EA6;&#x4E5F;&#x8D8A;&#x5927;&#xFF0C;<script type="math/tex; ">|\mathcal{H}|</script>&#x6709;&#x9650;&#x65F6;&#xFF0C;&#x6211;&#x4EEC;&#x79F0;<script type="math/tex; ">\mathcal{H}</script>&#x4E3A;&#x6709;&#x9650;&#x5047;&#x8BBE;&#x7A7A;&#x95F4;&#xFF0C;&#x5426;&#x5219;&#x79F0;&#x4E3A;&#x201C;&#x65E0;&#x9650;&#x5047;&#x8BBE;&#x7A7A;&#x95F4;&#x201D;&#x3002;</p>
</li>
<li><p>Page273: &#x4E0D;&#x53EF;&#x77E5;PAC&#x53EF;&#x5B66;&#x4E60;(agnostic PAC learnable)</p>
<p>  &#x4EE4;<script type="math/tex; ">m</script>&#x8868;&#x793A;&#x4ECE;&#x5206;&#x5E03;<script type="math/tex; ">\mathcal{D}</script>&#x4E2D;&#x72EC;&#x7ACB;&#x540C;&#x5206;&#x5E03;&#x91C7;&#x6837;&#x5F97;&#x5230;&#x7684;&#x6837;&#x4F8B;&#x6570;&#x76EE;&#xFF0C;<script type="math/tex; ">1<\epsilon, \delta < 1</script>&#xFF0C;&#x5BF9;&#x6240;&#x6709;&#x5206;&#x5E03;<script type="math/tex; ">\mathcal{D}</script>&#xFF0C;&#x82E5;&#x5B58;&#x5728;&#x5B66;&#x4E60;&#x7B97;&#x6CD5;<script type="math/tex; ">\mathcal{L}</script>&#x548C;&#x591A;&#x9879;&#x5F0F;&#x51FD;&#x6570;<script type="math/tex; ">poly(.,.,.,.)</script>&#xFF0C;&#x4F7F;&#x5F97;&#x5BF9;&#x4E8E;&#x4EFB;&#x4F55;<script type="math/tex; ">m\leqq poly(1/\epsilon, 1/\delta, size(x), size(c))</script>&#xFF0C;<script type="math/tex; ">\mathfrak{L}</script>&#x80FD;&#x4ECE;&#x5047;&#x8BBE;&#x7A7A;&#x95F4;<script type="math/tex; ">\mathcal{H}</script>&#x4E2D;&#x8F93;&#x51FA;&#x6EE1;&#x8DB3;<script type="math/tex; ">P(E(h)-min_{h^{'} \in \mathcal{H}})\leqq\epsilon\geqq 1 - \delta</script>&#x7684;&#x5047;&#x8BBE;h,&#x5219;&#x79F0;&#x5047;&#x8BBE;&#x7A7A;&#x95F4;<script type="math/tex; ">\mathcal{H}</script>&#x662F;&#x4E0D;&#x53EF;&#x77E5;PAC&#x53EF;&#x5B66;&#x4E60;&#x7684;&#x3002;</p>
</li>
<li><p>Page273: &#x589E;&#x957F;&#x51FD;&#x6570;</p>
<p>  &#x7ED9;&#x5B9A;&#x5047;&#x8BBE;&#x7A7A;&#x95F4;<script type="math/tex; ">\mathcal{H}</script>&#x548C;&#x793A;&#x4F8B;&#x96C6;<script type="math/tex; ">D={x_1, x_2, ..., x_m}</script>&#xFF0C;<script type="math/tex; ">\mathcal{H}</script>&#x4E2D;&#x6BCF;&#x4E2A;&#x5047;&#x8BBE;h&#x90FD;&#x80FD;&#x5BF9;<script type="math/tex; ">D</script>&#x4E2D;&#x793A;&#x4F8B;&#x8D4B;&#x4E88;&#x6807;&#x8BB0;&#xFF0C;&#x6807;&#x8BB0;&#x7ED3;&#x679C;&#x53EF;&#x8868;&#x793A;&#x4E3A;</p>
<p>  <script type="math/tex; ">h|_D = {h(x_1), h(x_2), ... , h(x_m)}</script></p>
<p>  &#x968F;&#x7740;m&#x7684;&#x589E;&#x5927;&#xFF0C;<script type="math/tex; ">\mathcal{H}</script>&#x4E2D;&#x6240;&#x6709;&#x5047;&#x8BBE;&#x5BF9;D&#x4E2D;&#x7684;&#x793A;&#x4F8B;&#x6240;&#x80FD;&#x8D4B;&#x4E88;&#x6807;&#x8BB0;&#x7684;&#x53EF;&#x80FD;&#x7ED3;&#x679C;&#x6570;&#x4E5F;&#x4F1A;&#x589E;&#x5927;&#x3002;</p>
<p>  &#x5BF9;&#x6240;&#x6709;<script type="math/tex; ">m\in\mathbb{N}</script>&#xFF0C;&#x5047;&#x8BBE;&#x7A7A;&#x95F4;<script type="math/tex; ">\mathcal{H}</script>&#x7684;&#x589E;&#x957F;&#x51FD;&#x6570;</p>
<p>  <script type="math/tex; ">\prod_{\mathcal{H}}(m)= max_{\{x_1, ..., x_m\subseteq \mathcal{H}\}}|\{h(x_1), h(x_2),..., h(x_m)|h\subseteq \mathcal{H}\}| </script>&#xFF0C;</p>
<p>  &#x8868;&#x793A;&#x5047;&#x8BBE;&#x7A7A;&#x95F4;<script type="math/tex; ">\mathcal{H}</script>&#x5BF9;m&#x4E2A;&#x793A;&#x4F8B;&#x6240;&#x80FD;&#x8D4B;&#x4E88;&#x6807;&#x8BB0;&#x7684;&#x6700;&#x5927;&#x53EF;&#x80FD;&#x7ED3;&#x679C;&#x6570;&#xFF0C;&#x53EF;&#x80FD;&#x7ED3;&#x679C;&#x6570;&#x8D8A;&#x5927;&#xFF0C;&#x5047;&#x8BBE;&#x7A7A;&#x95F4;&#x7684;&#x8868;&#x793A;&#x80FD;&#x529B;&#x8D8A;&#x5F3A;&#xFF0C;&#x5BF9;&#x5B66;&#x4E60;&#x4EFB;&#x52A1;&#x7684;&#x9002;&#x5E94;&#x80FD;&#x529B;&#x4E5F;&#x8D8A;&#x5F3A;&#xFF0C;&#x53EF;&#x4EE5;&#x5229;&#x7528;&#x589E;&#x957F;&#x51FD;&#x6570;&#x6765;&#x4F30;&#x8BA1;&#x7ECF;&#x9A8C;&#x8BEF;&#x5DEE;&#x4E0E;&#x6CDB;&#x5316;&#x8BEF;&#x5DEE;&#x4E4B;&#x95F4;&#x7684;&#x5173;&#x7CFB;&#x3002;</p>
</li>
</ul>
<ul>
<li><p>Page273: &#x5BF9;&#x5206;</p>
<p>  &#x5047;&#x8BBE;&#x7A7A;&#x95F4;<script type="math/tex; ">\mathcal{H}</script>&#x4E2D;&#x4E0D;&#x540C;&#x7684;&#x5047;&#x8BBE;&#x5BF9;&#x4E8E;<script type="math/tex; ">\mathcal{D}</script>&#x4E2D;&#x793A;&#x4F8B;&#x8D4B;&#x4E88;&#x6807;&#x8BB0;&#x7684;&#x7ED3;&#x679C;&#x53EF;&#x80FD;&#x76F8;&#x540C;&#xFF0C;&#x4E5F;&#x53EF;&#x80FD;&#x4E0D;&#x540C;&#xFF1B;&#x5C3D;&#x7BA1;<script type="math/tex; ">\mathcal{H}</script>&#x53EF;&#x80FD;&#x5305;&#x542B;&#x65E0;&#x7A77;&#x591A;&#x4E2A;&#x5047;&#x8BBE;&#xFF0C;&#x4F46;&#x5176;&#x5BF9;<script type="math/tex; ">\mathcal{D}</script>&#x4E2D;&#x793A;&#x4F8B;&#x8D4B;&#x4E88;&#x6807;&#x8BB0;&#x7684;&#x53EF;&#x80FD;&#x7ED3;&#x679C;&#x662F;&#x6709;&#x9650;&#x7684;&#x3002;&#x5BF9;m&#x4E2A;&#x793A;&#x4F8B;&#xFF0C;&#x6700;&#x591A;&#x6709;<script type="math/tex; ">2^m</script>&#x4E2A;&#x53EF;&#x80FD;&#x7ED3;&#x679C;&#xFF0C;&#x5BF9;&#x4E8C;&#x5206;&#x7C7B;&#x95EE;&#x9898;&#x6765;&#x8BF4;&#xFF0C;<script type="math/tex; ">\mathcal{H}</script>&#x4E2D;&#x7684;&#x5047;&#x8BBE;&#x5BF9;<script type="math/tex; ">\mathcal{D}</script>&#x4E2D;&#x793A;&#x4F8B;&#x8D4B;&#x4E88;&#x6807;&#x8BB0;&#x7684;&#x6BCF;&#x79CD;&#x53EF;&#x80FD;&#x7ED3;&#x679C;&#x79F0;&#x4E3A;&#x5BF9;<script type="math/tex; ">\mathcal{D}</script>&#x7684;&#x4E00;&#x79CD;&#x5BF9;&#x5206;&#x3002;</p>
</li>
<li><p>Page273: &#x6253;&#x6563;</p>
<p>  &#x82E5;&#x5047;&#x8BBE;&#x7A7A;&#x95F4;<script type="math/tex; ">\mathcal{H}</script>&#x80FD;&#x5B9E;&#x73B0;&#x793A;&#x4F8B;&#x96C6;<script type="math/tex; ">\mathcal{D}</script>&#x4E0A;&#x7684;&#x6240;&#x6709;&#x5BF9;&#x5206;&#xFF0C;&#x5373;<script type="math/tex; ">\prod_{\mathcal{H}}(m)=2^m</script>&#xFF0C;&#x5219;&#x79F0;&#x793A;&#x4F8B;&#x96C6;<script type="math/tex; ">\mathcal{D}</script>&#x80FD;&#x88AB;&#x5047;&#x8BBE;&#x7A7A;&#x95F4;<script type="math/tex; ">\mathcal{H}</script>&#x6253;&#x6563;&#x3002;</p>
</li>
<li><p>Page273: VC&#x7EF4;(274)</p>
<p>  &#x5047;&#x8BBE;&#x7A7A;&#x95F4;&#x7684;VC&#x7EF4;&#x662F;&#x80FD;&#x88AB;<script type="math/tex; ">\mathcal{H}</script>&#x6253;&#x6563;&#x7684;&#x6700;&#x5927;&#x793A;&#x4F8B;&#x96C6;&#x7684;&#x5927;&#x5C0F;&#xFF0C;&#x5373;<script type="math/tex; ">VC(\mathcal{H})=max{m: \prod_{\mathcal{H}}(m)=2^m}</script>&#xFF0C;&#x53EF;&#x7528;&#x4E8E;&#x5EA6;&#x91CF;&#x5047;&#x8BBE;&#x7A7A;&#x95F4;&#x7684;&#x590D;&#x6742;&#x5EA6;&#x3002;</p>
</li>
</ul>
<ul>
<li><p>Page278: &#x7ECF;&#x9A8C;&#x98CE;&#x9669;&#x6700;&#x5C0F;&#x5316;(Empirical Risk Minimization, ERM)</p>
<p>  &#x4EE4;h&#x8868;&#x793A;&#x5B66;&#x4E60;&#x7B97;&#x6CD5;<script type="math/tex; ">\mathfrak{L}</script>&#x8F93;&#x51FA;&#x7684;&#x5047;&#x8BBE;&#xFF0C;&#x82E5;<script type="math/tex; ">h</script>&#x6EE1;&#x8DB3;<script type="math/tex; ">\hat{E}(h)=min_{h'}</script>&#xFF0C;&#x5219;&#x79F0;<script type="math/tex; ">\mathfrak{L}</script>&#x4E3A;&#x6EE1;&#x8DB3;&#x7ECF;&#x9A8C;&#x98CE;&#x9669;&#x6700;&#x5C0F;&#x5316;&#x539F;&#x5219;&#x7684;&#x7B97;&#x6CD5;&#x3002;</p>
</li>
<li><p>Page279: Rademacher&#x590D;&#x6742;&#x5EA6;</p>
<p>  &#x7ECF;&#x9A8C;&#x8BEF;&#x5DEE;&#x6700;&#x5C0F;&#x7684;&#x5047;&#x8BBE;&#x662F;</p>
<p>  <script type="math/tex; ">argmax_{h\in\mathcal{H}}\frac{1}{m}\sum_{i=1}^m y_i h(x_i)</script>&#xFF0C;</p>
<p>  &#x7136;&#x800C;&#x73B0;&#x5B9E;&#x4EFB;&#x52A1;&#x4E2D;&#x7684;&#x6807;&#x8BB0;y&#x6709;&#x65F6;&#x5019;&#x4F1A;&#x6536;&#x5230;&#x566A;&#x58F0;&#x5F71;&#x54CD;&#xFF0C;&#x4E0D;&#x518D;&#x662F;&#x771F;&#x5B9E;&#x6807;&#x8BB0;&#xFF0C;&#x518D;&#x6B64;&#x60C5;&#x5F62;&#x4E0B;&#xFF0C;&#x9009;&#x62E9;&#x5047;&#x8BBE;&#x7A7A;&#x95F4;<script type="math/tex; ">\mathcal{H}</script>&#x4E2D;&#x5728;&#x8BAD;&#x7EC3;&#x96C6;&#x4E0A;&#x8868;&#x73B0;&#x6700;&#x597D;&#x7684;&#x5047;&#x8BBE;&#xFF0C;&#x6709;&#x65F6;&#x8FD8;&#x4E0D;&#x5982;&#x9009;&#x62E9;<script type="math/tex; ">\mathcal{H}</script>&#x4E2D;&#x4E8B;&#x5148;&#x8003;&#x8651;&#x4E86;&#x968F;&#x673A;&#x566A;&#x58F0;&#x5F71;&#x54CD;&#x7684;&#x5047;&#x8BBE;&#x3002;&#x8003;&#x8651;&#x968F;&#x673A;&#x53D8;&#x91CF;<script type="math/tex; ">\sigma_i</script>&#xFF0C;&#x5B83;&#x4EE5;0.5&#x7684;&#x6982;&#x7387;&#x53D6;&#x503C;-1,0.5&#x7684;&#x6982;&#x7387;&#x53D6;&#x503C;+1&#xFF0C;&#x79F0;&#x4E3A;Rademacher&#x968F;&#x673A;&#x53D8;&#x91CF;&#xFF0C;&#x57FA;&#x4E8E;<script type="math/tex; ">\sigma_i</script>&#x53EF;&#x5C06;&#x4E0A;&#x5F0F;&#x6539;&#x5199;&#x4E3A;</p>
<p>  <script type="math/tex; ">sup_{h\in\mathcal{H}}\frac{1}{m}\sum^m_{i=1}\sigma_i h(x_i)</script>,</p>
<p>  &#x671F;&#x671B;&#x4E3A;</p>
<p>  <script type="math/tex; ">\mathbb{E}_{\sigma}[sup_{h\in\mathcal{H}}\sum_{i=1}^m \sigma_ih(x_i)]</script>&#xFF0C;</p>
<p>  &#x8003;&#x8651;&#x5B9E;&#x503C;&#x51FD;&#x6570;&#x7A7A;&#x95F4;
  <script type="math/tex; ">\mathcal{F}:\mathcal{Z}\rightarrow\mathcal{R}</script>&#xFF0C;
  &#x4EE4;<script type="math/tex; ">Z={z_1,z_2,...,z_m}</script>&#xFF0C;</p>
<p>  &#x5176;&#x4E2D;<script type="math/tex; ">z_i\in\mathcal{Z}</script>, &#x5C06;<script type="math/tex; ">\mathcal{X}</script>&#x548C;<script type="math/tex; ">\mathcal{H}</script>&#x66FF;&#x6362;&#x6210;<script type="math/tex; ">Z</script>&#x548C;<script type="math/tex; ">F</script>&#x53EF;&#x5F97;&#xFF0C;&#x51FD;&#x6570;&#x7A7A;&#x95F4;<script type="math/tex; ">\mathcal{Z}</script>g&#x5173;&#x4E8E;<script type="math/tex; ">\mathcal{F}</script>&#x7684;&#x7ECF;&#x9A8C;Rademacher&#x590D;&#x6742;&#x5EA6;</p>
<p>  <script type="math/tex; ">\hat{R}_Z(\mathcal{F})=\mathbb{E}_{\sigma}[sup_{f\in\mathcal{F}}\frac{1}{m}\sum_{i=1}^m\sigma_i f(z_i)]</script>&#xFF0C;</p>
<p>  &#x5176;&#x8861;&#x91CF;&#x4E86;&#x51FD;&#x6570;&#x7A7A;&#x95F4;<script type="math/tex; ">\mathcal{F}</script>&#x4E0E;&#x968F;&#x673A;&#x566A;&#x58F0;&#x5728;&#x96C6;&#x5408;<script type="math/tex; ">\mathcal{Z}</script>&#x4E2D;&#x7684;&#x76F8;&#x5173;&#x6027;&#x3002;&#x5BF9;&#x6240;&#x6709;&#x4ECE;<script type="math/tex; ">\mathcal{D}</script>&#x72EC;&#x7ACB;&#x540C;&#x5206;&#x5E03;&#x91C7;&#x6837;&#x800C;&#x5F97;&#x7684;&#x5927;&#x5C0F;&#x4E3A;m&#x7684;&#x96C6;&#x5408;Z&#x6C42;&#x671F;&#x671B;&#x53EF;&#x5F97;&#x51FD;&#x6570;&#x7A7A;&#x95F4;<script type="math/tex; ">\mathcal{F}</script>&#x5173;&#x4E8E;&#x4E0A;<script type="math/tex; ">\mathcal{Z}</script>&#x5206;&#x5E03;&#x7684;<script type="math/tex; ">\mathcal{D}</script>Rademacher&#x590D;&#x6742;&#x5EA6;&#x3002;</p>
</li>
<li><p>Page284: &#x7A33;&#x5B9A;&#x6027;</p>
<p>  &#x7B97;&#x6CD5;&#x5728;&#x8F93;&#x5165;&#x53D1;&#x751F;&#x53D8;&#x5316;&#x65F6;&#xFF0C;&#x8F93;&#x51FA;&#x662F;&#x5426;&#x4F1A;&#x968F;&#x4E4B;&#x53D1;&#x751F;&#x8F83;&#x5927;&#x7684;&#x53D8;&#x5316;&#x3002;</p>
</li>
<li><p>Page285: &#x5747;&#x5300;&#x7A33;&#x5B9A;&#x6027;</p>
<p>  &#x5BF9;&#x4EFB;&#x4F55;<script type="math/tex; ">x\in\mathcal{X},z=(x,y)</script>&#xFF0C;&#x82E5;&#x5B66;&#x4E60;&#x7B97;&#x6CD5;<script type="math/tex; ">\mathcal{L}</script>&#x6EE1;&#x8DB3;</p>
<p>  <script type="math/tex; ">|l(\mathcal{L}_D, z) - l(\mathcal{L}_{D_i},z)|\leqq\beta</script>&#xFF0C;</p>
<p>  &#x5219;&#x79F0;<script type="math/tex; ">\mathcal{L}</script>&#x5173;&#x4E8E;&#x635F;&#x5931;&#x51FD;&#x6570;<script type="math/tex; ">l</script>&#x6EE1;&#x8DB3;<script type="math/tex; ">\beta</script>&#x5747;&#x5300;&#x7A33;&#x5B9A;&#x6027;&#x3002;</p>
<p>  &#x5176;&#x4E2D;l&#x4E3A;&#x6CDB;&#x5316;&#x635F;&#x5931;&#xFF0C;<script type="math/tex; ">D_i</script>&#x4E3A;&#x79FB;&#x9664;<script type="math/tex; ">D</script>&#x4E2D;&#x7B2C;i&#x4E2A;&#x6837;&#x4F8B;&#x5F97;&#x5230;&#x7684;&#x96C6;&#x5408;&#x3002;</p>
</li>
</ul>

                                
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